Biology subjects

Simpson, M. J.

Publications and source records attributed to Simpson, M. J..

At least 37 records · Page 2Linked to original sources

bioRxiv2021

A stochastic mathematical model of 4D tumour spheroids with real-time fluorescent cell cycle labelling

In vitro tumour spheroid experiments have been used to study avascular tumour growth and drug design for the last 50 years. Unlike simpler two-dimensional cell cultures, tumour spheroids exhibit heterogeneity within the growing population of cells that is thought to be related to spatial and temporal differences in nutrient availability. The recent development of real-time fluorescent cell cycle imaging allows us to identify the position and cell cycle status of individual cells within the growing population, giving rise to the notion of a four-dimensional (4D) tumour spheroid. In this work we develop the first stochastic individual-based model (IBM) of a 4D tumour spheroid and show that IBM simulation data qualitatively and quantitatively compare very well with experimental data from a suite of 4D tumour spheroid experiments performed with a primary human melanoma cell line. The IBM provides quantitative information about nutrient availability within the spheroid, which is important because it is very difficult to measure these data in standard tumour spheroid experiments. Software required to implement the IBM is available on GitHub, https://github.com/ProfMJSimpson/4DFUCCI.

biophysics↗

bioRxiv2021

Identifying cell-to-cell variability in internalisation using flow cytometry

Biological heterogeneity is a primary contributor to the variation observed in experiments that probe dynamical processes, such as internalisation. Given that internalisation is a critical process by which many therapeutics and viruses reach their intracellular site of action, quantifying cell-to-cell variability in internalisation is of high biological interest. Yet, it is common for studies of internalisation to neglect cell-to-cell variability. We develop a simple mathematical model of internalisation that captures the dynamical behaviour, cell-to-cell variation, and extrinsic noise introduced by flow cytometry. We calibrate our model through a novel distribution-matching approximate Bayesian computation algorithm to flow cytometry data of internalisation of anti-transferrin receptor antibody in a human B-cell lymphoblastoid cell line. Our model reproduces experimental observations, identifies cell-to-cell variability in the internalisation and recycling rates, and, importantly, provides information relating to inferential uncertainty. Given that our approach is agnostic to sample size and signal-to-noise ratio, our modelling framework is broadly applicable to identify biological variability in single-cell data from internalisation assays and similar experiments that probe cellular dynamical processes.

cell biology↗

bioRxiv2021

Designing and interpreting 4D tumour spheroid experiments

Tumour spheroid experiments are routinely used to study cancer progression and treatment. Various and inconsistent experimental designs are used, leading to challenges in interpretation and reproducibility. Using multiple experimental designs, live-dead cell staining, and real-time cell cycle imaging, we measure necrotic and proliferation-inhibited regions in over 1000 4D tumour spheroids (3D space plus cell cycle status). By intentionally varying the initial spheroid size and temporal sampling frequencies across multiple cell lines, we collect an abundance of measurements of internal spheroid structure. These data are difficult to compare and interpret. However, using an objective mathematical modelling framework and statistical identifiability analysis we quantitatively compare experimental designs and identify design choices that produce reliable biological insight. Measurements of internal spheroid structure provide the most insight, whereas varying initial spheroid size and temporal measurement frequency is less important. Our general framework applies to spheroids grown in different conditions and with different cell types.

biophysics↗

bioRxiv2021

Quantitative analysis of tumour spheroid structure

Tumour spheroids are common in vitro experimental models of avascular tumour growth. Compared with traditional two-dimensional culture, tumour spheroids more closely mimic the avascular tumour microenvironment where spatial differences in nutrient availability strongly influence growth. We show that spheroids initiated using significantly different numbers of cells grow to similar limiting sizes, suggesting that avascular tumours have a limiting structure; in agreement with untested predictions of classical mathematical models of tumour spheroids. We develop a novel mathematical and statistical framework to study the structure of tumour spheroids seeded from cells transduced with fluorescent cell cycle indicators, enabling us to discriminate between arrested and cycling cells and identify an arrested region. Our analysis shows that transient spheroid structure is independent of initial spheroid size, and the limiting structure can be independent of seeding density. Standard experimental protocols compare spheroid size as a function of time; however, our analysis suggests that comparing spheroid structure as a function of overall size produces results that are relatively insensitive to variability in spheroid size. Our experimental observations are made using two melanoma cell lines, but our modelling framework applies across a wide range of spheroid culture conditions and cell lines.

cancer biology↗

bioRxiv2021

Parameter identifiability and model selection for sigmoid population growth models

Sigmoid growth models, such as the logistic, Gompertz and Richards models, are widely used to study population dynamics ranging from microscopic populations of cancer cells, to continental-scale human populations. Fundamental questions about model selection and parameter estimation are critical if these models are to be used to make practical inferences. However, the question of parameter identifiability - whether a data set contains sufficient information to give unique or sufficiently precise parameter estimates - is often overlooked. We use a profile-likelihood approach to explore practical parameter identifiability using data describing the re-growth of hard coral. With this approach, we explore the relationship between parameter identifiability and model misspecification, finding that the logistic growth model does not suffer identifiability issues for the type of data we consider whereas the Gompertz and Richards models encounter practical non-identifiability issues. This analysis of parameter identifiability and model selection is important because different growth models are used within areas of the biological modelling literature without necessarily considering whether parameters are identifiable, or checking statistical assumptions underlying model adequacy. Standard practices that do not consider parameter identifiability can lead to unreliable or imprecise parameter estimates and potentially misleading mechanistic interpretations. While tools developed here focus on three standard sigmoid growth models only, our theoretical developments are applicable to any sigmoid growth model and any relevant data set. MATLAB implementations of all software are available on GitHub.

ecology↗

bioRxiv2021

Estimating parameters of a stochastic cell invasion model with fluorescent cell cycle labelling using Approximate Bayesian Computation

We develop a parameter estimation method based on approximate Bayesian computation (ABC) for a stochastic cell invasion model using fluorescent cell cycle labeling with proliferation, migration, and crowding effects. Previously, inference has been performed on a deterministic version of the model fitted to cell density data, and not all the parameters were identifiable. Considering the stochastic model allows us to harness more features of experimental data, including cell trajectories and cell count data, which we show overcomes the parameter identifiability problem. We demonstrate that, whilst difficult to collect, cell trajectory data can provide more information about the parameters of the cell invasion model. To handle the intractability of the likelihood function of the stochastic model, we use an efficient ABC algorithm based on sequential Monte Carlo. Rcpp and MATLAB implementations of the simulation model and ABC algorithm used in this study are available at https://github.com/michaelcarr-stats/FUCCI.

cell biology↗

bioRxiv2021

Model-based data analysis of tissue growth in thin 3D printed scaffolds

Tissue growth in three-dimensional (3D) printed scaffolds enables exploration and control of cell behaviour in biologically realistic geometries. Cell proliferation and migration in these experiments have yet to be explicitly characterised, limiting the ability of experimentalists to determine the effects of various experimental conditions, such as scaffold geometry, on cell behaviour. We consider tissue growth by osteoblastic cells in melt electro-written scaffolds that comprise thin square pores with sizes that we deliberately vary. We collect highly detailed temporal measurements of the average cell density, tissue coverage, and tissue geometry. To quantify tissue growth in terms of the underlying cell proliferation and migration processes, we introduce and calibrate a mechanistic mathematical model based on the Porous-Fisher reaction-diffusion equation. Parameter estimates and uncertainty quantification through profile likelihood analysis reveal consistency in the rate of cell proliferation and steady-state cell density between pore sizes. This analysis also serves as an important model verification tool: while the use of reaction-diffusion models in biology is widespread, the appropriateness of these models to describe tissue growth in 3D scaffolds has yet to be explored. We find that the Porous-Fisher model is able to capture features relating to the cell density and tissue coverage, but is not able to capture geometric features relating to the circularity of the tissue interface. Our analysis identifies two distinct stages of tissue growth, suggests several areas for model refinement, and provides guidance for future experimental work that explores tissue growth in 3D printed scaffolds. Author SummaryAdvances in 3D printing technology have led to cell culture experiments that realistically capture natural biological environments. Despite the necessity of quantifying cell behaviour with parameters that can be compared between experiments, many existing mathematical models of tissue growth in these experiments neglect information relating to population size. We consider tissue growth by cells on 3D printed scaffolds that comprise square pores of various sizes in this work. We apply a relatively simple mathematical model based on the Porous-Fisher reaction-diffusion equation to interpret highly detailed measurements relating to both the cell density and the quantity of tissue deposited. We analyse the efficacy of such a model in capturing cell behaviour seen in the experiments and quantify cell behaviour in terms of parameters that carry a biologically meaningful interpretation. Our analysis identifies important areas for model refinement and provides guidance for future data-collection and experimentation that explores tissue growth in 3D printed scaffolds.

cell biology↗

bioRxiv2021

Implementation and acceleration of optimal control for systems biology

Optimal control theory provides insight into complex resource allocation decisions. The forward-backward sweep method (FBSM) is an iterative technique commonly implemented to solve two-point boundary value problems (TPBVPs) arising from the application of Pontryagins Maximum Principle (PMP) in optimal control. In this review we discuss the PMP approach to optimal control and the implementation of the FBSM. By conceptualising the FBSM as a fixed point iteration process, we leverage and adapt existing acceleration techniques to improve its rate of convergence. We show that convergence improvement is attainable without prohibitively costly tuning of the acceleration techniques. Further, we demonstrate that these methods can induce convergence where the underlying FBSM fails to converge. All code used in this work to implement the FBSM and acceleration techniques is available on GitHub at https://github.com/Jesse-Sharp/Sharp2021.

systems biology↗

bioRxiv2020

The role of mechanical interactions in EMT

The detachment of cells from the boundary of an epithelial tissue and the subsequent invasion of these cells into surrounding tissues is important for cancer development and wound healing, and is strongly associated with the epithelial-mesenchymal transition (EMT). Chemical signals, such as TGF-{beta}, produced by surrounding tissue can be up-taken by cells and induce EMT. In this work, we present a novel cell-based discrete mathematical model of mechanical cellular relaxation, cell proliferation, and cell detachment driven by chemically-dependent EMT in an epithelial tissue. A continuum description of the model is then derived in the form of a novel nonlinear free boundary problem. Using the discrete and continuum models we explore how the coupling of chemical transport and mechanical interactions influences EMT, and postulate how this could be used to help control EMT in pathological situations.

biophysics↗

bioRxiv2020

Mathematical model of tumour spheroid experiments with real-time cell cycle imaging

Three-dimensional (3D) in vitro tumour spheroid experiments are an important tool for studying cancer progression and potential drug therapies. Standard experiments involve growing and imaging spheroids to explore how different experimental conditions lead to different rates of spheroid growth. These kinds of experiments, however, do not reveal any information about the spatial distribution of the cell cycle within the expanding spheroid. Since 2008, a new experimental technology called fluorescent ubiquitination-based cell cycle indicator (FUCCI), has enabled real time in situ visualisation of the cell cycle progression. FUCCI labelling involves cells in G1 phase of the cell cycle fluorescing red, and cells in the S/G2/M phase of the cell cycle fluorescing green. Experimental observations of 3D tumour spheroids with FUCCI labelling reveal significant intratumoural structure, as the cell cycle status can vary with location. Although many mathematical models of tumour spheroid growth have been developed, none of the existing mathematical models are designed to interpret experimental observations with FUCCI labelling. In this work we extend the mathematical framework originally proposed by Ward and King (1997) to develop a new mathematical model of FUCCI-labelled tumour spheroid growth. The mathematical model treats the spheroid as being composed of three subpopulations: (i) living cells in G1 phase that fluoresce red; (ii) living cells in S/G2/M phase that fluoresce green; and, (iii) dead cells that do not fluoresce. We assume that the rates at which cells pass through different phases of the cell cycle, and the rate of cell death, depend upon the local oxygen concentration in the spheroid. Parameterising the new mathematical model using experimental measurements of cell cycle transition times, we show that the model can capture important experimental observations that cannot be addressed using previous mathematical models. Further, we show that the mathematical model can be used to quantitatively mimic the action of anti-mitotic drugs applied to the spheroid. All software required to solve the nonlinear moving boundary problem associated with the new mathematical model are available on GitHub.

cancer biology↗

bioRxiv2020

Listen to the noise: identifiability analysis for stochastic differential equation models in systems biology

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly-adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically-motivated synthetic data and Markov-chain Monte Carlo (MCMC) methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

systems biology↗

bioRxiv2020

A free boundary mechanobiological model of epithelial tissues

In this study, we couple intracellular signalling and cell-based mechanical properties to develop a novel free boundary mechanobiological model of epithelial tissue dynamics. Mechanobiological coupling is introduced at the cell level in a discrete modelling framework, and new reaction-diffusion equations are derived to describe tissue-level outcomes. The free boundary evolves as a result of the underlying biological mechanisms included in the discrete model. To demonstrate the accuracy of the continuum model, we compare numerical solutions of the discrete and continuum models for two different signalling pathways. First, we study the Rac-Rho pathway where cell- and tissue-level mechanics are directly related to intracellular signalling. Second, we study an activator-inhibitor system which gives rise to spatial and temporal patterning related to Turing patterns. In all cases, the continuum model and free boundary condition accurately reflect the cell-level processes included in the discrete model.

biophysics↗

bioRxiv2020

Population dynamics with spatial structure and an Allee effect

Allee effects describe populations in which long-term survival is only possible if the population density is above some threshold level. A simple mathematical model of an Allee effect is one where initial densities below the threshold lead to population extinction, whereas initial densities above the threshold eventually asymptote to some positive carrying capacity density. Mean field models of population dynamics neglect spatial structure that can arise through short-range interactions, such as short-range competition and dispersal. The influence of such non mean-field effects has not been studied in the presence of an Allee effect. To address this we develop an individual-based model (IBM) that incorporates both short-range interactions and an Allee effect. To explore the role of spatial structure we derive a mathematically tractable continuum approximation of the IBM in terms of the dynamics of spatial moments. In the limit of long-range interactions where the mean-field approximation holds, our modelling framework accurately recovers the mean-field Allee threshold. We show that the Allee threshold is sensitive to spatial structure that mean-field models neglect. For example, we show that there are cases where the mean-field model predicts extinction but the population actually survives and vice versa. Through simulations we show that our new spatial moment dynamics model accurately captures the modified Allee threshold in the presence of spatial structure.

biophysics↗

bioRxiv2020

Population dynamics with threshold effects give rise to a diverse family of Allee effects

The Allee effect describes populations that deviate from logistic growth models and arises in applications including ecology and cell biology. A common justification for incorporating Allee effects into population models is that the population in question has altered growth mechanisms at some critical density, often referred to as a threshold effect. Despite the ubiquitous nature of threshold effects arising in various biological applications, the explicit link between local threshold effects and global Allee effects has not been considered. In this work, we examine a continuum population model that incorporates threshold effects in the local growth mechanisms. We show that this model gives rise to a diverse family of Allee effects and we provide a comprehensive analysis of which choices of local growth mechanisms give rise to specific Allee effects. Calibrating this model to a recent set of experimental data describing the growth of a population of cancer cells provides an interpretation of the threshold population density and growth mechanisms associated with the population.

biophysics↗

bioRxiv2020

Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size

Tissue growth in bioscaffolds is influenced significantly by pore geometry, but how this geometric dependence emerges from dynamic cellular processes such as cell proliferation and cell migration remains poorly understood. Here we investigate the influence of pore size on the time required to bridge pores in thin 3D-printed scaffolds. Experimentally, new tissue infills the pores continually from their perimeter under strong curvature control, which leads the tissue front to round off with time. Despite the varied shapes assumed by the tissue during this evolution, we find that time to bridge a pore simply increases linearly with the overall pore size. To disentangle the biological influence of cell behaviour and the mechanistic influence of geometry in this experimental observation, we propose a simple reaction-diffusion model of tissue growth based on Porous-Fisher invasion of cells into the pores. First, this model provides a good qualitative representation of the evolution of the tissue; new tissue in the model grows at an effective rate that depends on the local curvature of the tissue substrate. Second, the model suggests that a linear dependence of bridging time with pore size arises due to geometric reasons alone, not to differences in cell behaviours across pores of different sizes. Our analysis suggests that tissue growth dynamics in these experimental constructs is dominated by mechanistic crowding effects that influence collective cell proliferation and migration processes, and that can be predicted by simple reaction-diffusion models of cells that have robust, consistent behaviours.

bioengineering↗

bioRxiv2020

Synchronised oscillations in growing cell populations are explained by demographic noise

Understanding synchrony in growing populations is important for applications as diverse as epidemiology and cancer treatment. Recent experiments employing fluorescent reporters in melanoma cell lines have uncovered growing subpopulations exhibiting sustained oscillations, with nearby cells appearing to synchronise their cycles. In this study we demonstrate that the behaviour observed is consistent with long-lasting transient phenomenon initiated, and amplified by the finite-sample effects and demographic noise. We present a novel mathematical analysis of a multi-stage model of cell growth which accurately reproduces the synchronised oscillations. As part of the analysis, we elucidate the transient and asymptotic phases of the dynamics and derive an analytical formula to quantify the effect of demographic noise in the appearance of the oscillations. The implications of these findings are broad, such as providing insight into experimental protocols that are used to study the growth of asynchronous populations and, in particular, those investigations relating to anti-cancer drug discovery. Statement of SignificanceRecent experiments have reported strong evidence of periodic oscillations in the proportion of young and old melanoma cells. The biological mechanism generating this synchronisation and the potential impact that can have on commonly used experimental protocols is still unclear. Here we studied a population of melanoma cells for which we found oscillations in the proportions of cells in each phase of the cell cycle. We demonstrate that these observations may be triggered by intrinsic demographic noise alone, rather than any active synchronisation mechanism requiring cell-cell communication. Our findings may have implications for typical experimental protocols which aim to produce asynchronous cell populations.

cell biology↗

bioRxiv2020

New mathematical modelling tools for co-culture experiments: when do we need to explicitly account for signalling molecules?

Mathematical models are often applied to describe cell migration regulated by diffusible signalling molecules. A typical feature of these models is that the spatial and temporal distribution of the signalling molecule density is reported by solving a reaction-diffusion equation. However, the spatial and temporal distributions of such signalling molecules are not often reported or observed experimentally. This leads to a mismatch between the amount of experimental data available and the complexity of the mathematical model used to simulate the experiment. To address this mismatch, we develop a discrete model of cell migration that can be used to describe a new suite of co-culture cell migration assays involving two interacting subpopulations of cells. In this model, the migration of cells from one subpopulation is regulated by the presence of signalling molecules that are secreted by the other subpopulation of cells. The spatial and temporal distribution of the signalling molecules is governed by a discrete conservation statement that is related to a reaction-diffusion equation. We simplify the model by invoking a steady state assumption for the diffusible molecules, leading to a reduced discrete model allowing us to describe how one subpopulation of cells stimulates the migration of the other subpopulation of cells without explicitly dealing with the diffusible molecules. We provide additional mathematical insight into these two stochastic models by deriving continuum limit partial differential equation descriptions of both models. To understand the conditions under which the reduced model is a good approximation of the full model, we apply both models to mimic a set of novel co-culture assays and we systematically explore how well the reduced model approximates the full model as a function of the model parameters.

biophysics↗

bioRxiv2019

Persistence is an optimal hedging strategy for bacteria in volatile environments

Bacteria invest in a slow-growing subpopulation, called persisters, to ensure survival in the face of uncertainty. This hedging strategy is remarkably similar to financial hedging, where diversifying an investment portfolio protects against economic uncertainty. We provide a new theoretical foundation for understanding cellular hedging by unifying the study of biological population dynamics and the mathematics of financial risk management through optimal control theory. Motivated by the widely accepted role of volatility in the emergence of persistence, we consider several novel models of environmental volatility described by continuous-time stochastic processes. This allows us to study an emergent cellular hedging strategy that maximizes the expected per-capita growth rate of the population. Analytical and simulation results probe the optimal persister strategy, revealing results that are consistent with experimental observations and suggest at new opportunities for experimental investigation and design. Overall, we provide a new way of conceptualising and modelling cellular decision-making in volatile environments by explicitly unifying theory from mathematical biology and finance.

systems biology↗